## THE FARTHEST POINT PROBLEM IN NORMED SPACES

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##### Date

2023-03-13

##### Authors

Angham Khabisa

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##### Abstract

In this study, the researcher discusses the longstanding problem in Approximation Theory, which is called the Farthest Point Problem (FPP).
The FPP is partially an unsolved problem asking whether every uniquely remotal set in a Banach/Metric space is a singleton.
The researcher considers the convex metric space and demonstrates that every bounded subset is singleton if and only if SF condition is satisfied. Also, the researcher focuses on the Banach spaces. Firstly, in normed space the researcher proved that the singletoness occurs if partially continuous condition of farthest point map P satisfies. Then, the researcher takes a specific space in Banach space, the sequence space ℓ1(R) and shows the positive answer that every uniquely remotal subset of ℓ1(R) is singleton.
Finally, the researcher presents partially ideal statistically continuous notion of far- thest point map P and presents the main result, provided that E is a bounded, uniquely remotal set in a Banach space X over R with a Chebyshev center c and the farthest point map P defined on [c, P (c)] is partially ideal statistically continuous at c, then E consists of one element only.
Keyword: A Banach/Metric Space; Approximation Theory; Convex Metric Space; Partially Ideal Statistically Continuous